Improvement in accuracy for dimensionality reduction and reconstruction of noisy signals. Part I: The case of random signals

Our work addresses an improvement in accuracy for dimensionality reduction and reconstruction of random signals. The proposed transform targets noisy signals. This is because in the case of highly noisy signals, the known optimal methods might produce a large associated error. Let x, y and u be a source signal with m components, observed noisy signal with n components and reduced signal with k components, respectively, and let c=k/min{m,n} be a reduction ratio (RR) where k ≤ min {m, n}. If RR is fixed then the error associated with the known methods cannot be improved. The purpose of this paper is the development of an approach which leads to the better performance than that of the well-known fundamental techniques. The associated accuracy is improved by an optimal choice of an “auxiliary” random signal v (called the injection) and its dimensionality q, and by the increase in the number of matrices to optimize compared to the known transforms. At the same time, the total number of entries of the matrices is less than for the known related transforms. The proposed method contains, in particular, a special operation Q which transforms the large covariance matrix to the block diagonal form with smaller blocks. The above circumstances make the proposed technique numerically faster than the known related transforms. Numerical examples are provided that illustrate the above advantages.